MAXIMUM LIKELIHOOD ESTIMATION FOR AUTOREGRESSIVE PROCESSES DISTURBED BY A MOVING AVERAGE

Abstract. Maximum likelihood estimation for stationary autoregressive processes when the signal is subject to a moving-average sampling error is discussed. A modified maximum likelihood estimator is proposed. An algorithm for computing derivatives of the modified likelihood is suggested. Maximum likelihood estimators of the parameter vector are shown to be strongly consistent and to have a multivariate normal limiting distribution. A Monte Carlo simulation shows that the modified maximum likelihood estimator performs better than other available estimators. US current labour force data are analysed as an example.     

Robustness of Zero Crossing Estimator

Zero crossing (ZC) statistic is the number of zero crossings observed in a time series. The expected value of the ZC specifies the first‐order autocorrelation of the processes. Hence, we can estimate the autocorrelation by using the ZC estimator. The asymptotic consistency and normality of the ZC estimator for scalar Gaussian processes are already discussed in 1980. In this article, first, we derive the joint asymptotic distribution of the ZC estimator for ellipsoidal processes. Next, we show the variance of the ZC estimator does not attain the Cramer–Rao lower bound (CRLB). However, it is shown that the ZC estimator has robustness when the process is contaminated by an outlier. In contrast with this, we observe that the quasi‐maximum likelihood estimator (QMLE) attains the CRLB. However, we can see that QMLE is sensitive for the outlier.     

On the limit theory of the Gaussian SQMLE in the EGARCH(1,1) model

We derive the limit theory of the Gaussian stable quasi maximum likelihood estimator for the stationary EGARCH(1,1) model when the squared innovation process has marginals with regularly varying tails. We derive regularly varying rates and limiting stable distributions. We perform Monte Carlo experiments to assess the extent of the parameter space corresponding to the invertibility condition, and the quality of the asymptotic approximation.     

Transformation to approximate independence for locally stationary Gaussian processes

We provide new approximations for the likelihood of a time series under the locally stationary Gaussian process model. The likelihood approximations are valid even in cases when the evolutionary spectrum is not smooth in the rescaled time domain. We describe a broad class of models for the evolutionary spectrum for which the approximations can be computed particularly efficiently. In developing the approximations, we extend to the locally stationary case the idea that the discrete Fourier transform is a decorrelating transformation for stationary time series. The approximations are applied to fit non‐stationary time‐series models to high‐frequency temperature data. For these data, we fit evolutionary spectra that are piecewise constant in time and use a genetic algorithm to search for the best partition of the time interval.     

Likelihood analysis of a first-order autoregressive model with exponential innovations

Abstract. This paper derives the exact distribution of the maximum likelihood estimator of a first-order linear autoregression with an exponential disturbance term. We also show that, even if the process is stationary, the estimator is T -consistent, where T is the sample size. In the unit root case, the estimator is T ²-consistent, while, in the explosive case, the estimator is ρ ^T -consistent. Further, the likelihood ratio test statistic for a simple hypothesis on the autoregressive parameter is asymptotically uniform for all values of the parameter.     

Consistent Estimation of the Value at Risk When the Error Distribution of the Volatility Model is Misspecified

A two‐step approach for conditional value at risk estimation is considered. First, a generalized quasi‐maximum likelihood estimator is employed to estimate the volatility parameter, then the empirical quantile of the residuals serves to estimate the theoretical quantile of the innovations. When the instrumental density h of the generalized quasi‐maximum likelihood estimator is not the Gaussian density, both the estimations of the volatility and of the quantile are generally asymptotically biased. However, the two errors counterbalance and lead to a consistent estimator of the value at risk. We obtain the asymptotic behavior of this estimator and show how to choose optimally h.     

UNIFIED INTERVAL ESTIMATION FOR RANDOM COEFFICIENT AUTOREGRESSIVE MODELS

The consistency of the quasi‐maximum likelihood estimator for random coefficient autoregressive models requires that the coefficient be a non‐degenerate random variable. In this article, we propose empirical likelihood methods based on weighted‐score equations to construct a confidence interval for the coefficient. We do not need to distinguish whether the coefficient is random or deterministic and whether the process is stationary or non‐stationary, and we present two classes of equations depending on whether a constant trend is included in the model. A simulation study confirms the good finite‐sample behaviour of our resulting empirical likelihood‐based confidence intervals. We also apply our methods to study US macroeconomic data.     

Estimation of GARCH Models from the Autocorrelations of the Squares of a Process

This paper shows how the parameters of a stable GARCH(1, 1) model can be estimated from the autocorrelations of the squared process. Specifically, the method applies a minimum distance estimator (MDE) to the sample autocorrelations of the squared realization. The asymptotic efficiency of the estimator is calculated from using the first g autocorrelations. The estimator can be surprisingly efficient for quite small numbers of autocorrelations and, in some cases, can be more efficient than the quasi maximum likelihood estimator (QMLE). Also, the estimated process can better fit the pattern of observed autocorrelations of squared returns than those from models estimated by maximum likelihood estimation (MLE). The estimator is applied to a series of hourly exchange rate returns, which are extremely non Gaussian.     

STATIONARY TANGENT: THE DISCRETE AND NON-SMOOTH CASE

Abstract. In Keich (2000 ),we define a stationary tangent process, or a locally optimal stationary approximation, to a real non-stationary smooth Gaussian process. This paper extends the idea by constructing a discrete tangent – a `locally' optimal stationary approximation – for a discrete time, real Gaussian process. Analogously to the smooth case, our construction relies on a generalization of the recursion formula for the orthogonal polynomials of the spectral distribution function. More precisely, we use a generalization of the Schur parameters to identify the stationary tangent. By way of discretizing, we later demonstrate how this tangent can be used to obtain `good' local stationary approximations to non-smooth continuous time, real Gaussian processes. Further, we demonstrate how, analogously to the curvatures in the smooth case, the Schur parameters can be used to determine the order of stationarity of a non-smooth process.     

Some Notes on Mutual Information Between Past and Future

Abstract. We present some new results on the mutual information between past and future for Gaussian stationary sequences. We provide several formulae to calculate this quantity. As a by‐product, we establish the so‐called reflectrum identity that links partial autocorrelation coefficients and cepstrum coefficients. So as to obtain these results, we provide an account of several regularity conditions for Gaussian stationary processes in terms of properties of the associated Toeplitz and Hankel operators. We discuss conditions under which the mutual information is finite. These results lead us to an interesting perspective towards the definition of long‐memory processes. Our result implies that zeros on the unit circle can cause mutual information to be infinite. Examples include fractional autoregressive integrated moving average (ARIMA) models. In addition, we consider a finite sample from a Gaussian stationary sequence. In the expansion of the determinant of its covariance matrix, the Toeplitz matrix, the first and second term are, entropy and mutual information respectively. A form of approximation to the likelihood using entropy and mutual information is presented.     

Gaussian Maximum Likelihood Estimation For ARMA Models. I. Time Series

Abstract. We provide a direct proof for consistency and asymptotic normality of Gaussian maximum likelihood estimators for causal and invertible autoregressive moving‐average (ARMA) time series models, which were initially established by Hannan [Journal of Applied Probability (1973) vol. 10, pp. 130–145] via the asymptotic properties of a Whittle's estimator. This also paves the way to establish similar results for spatial processes presented in the follow‐up article by Yao and Brockwell [Bernoulli (2006) in press].     

Empirical Characteristic Functions-Based Estimation and Distance Correlation for Locally Stationary Processes

In this article, we propose a kernel-type estimator for the local characteristic function of locally stationary processes. Under weak moment conditions, we prove joint asymptotic normality for local empirical characteristic functions. For time-varying linear processes, we establish a central limit theorem under the assumption of finite absolute first moments of the process. Additionally, we prove weak convergence of the local empirical characteristic process. We apply our asymptotic results to parameter estimation. Furthermore, by extending the notion of distance correlation to locally stationary processes, we are able to provide asymptotic theory for local empirical distance correlations. Finally, we provide a simulation study on minimum distance estimation for α-stable distributions and illustrate the pairwise dependence structure over time of log returns of German stock prices via local empirical distance correlations.     

Efficient use of higher-lag autocorrelations for estimating autoregressive processes

The Yule–Walker estimator is commonly used in time-series analysis, as a simple way to estimate the coefficients of an autoregressive process. Under strong assumptions on the noise process, this estimator possesses the same asymptotic properties as the Gaussian maximum likelihood estimator. However, when the noise is a weak one, other estimators based on higher-order empirical autocorrelations can provide substantial efficiency gains. This is illustrated by means of a first-order autoregressive process with a Markov-switching white noise. We show how to optimally choose a linear combination of a set of estimators based on empirical autocorrelations. The asymptotic variance of the optimal estimator is derived. Empirical experiments based on simulations show that the new estimator performs well on the illustrative model.     

SOME PROPERTIES OF THE MAXIMUM LIKELIHOOD ESTIMATOR IN THE SIMULTANEOUS SWITCHING AUTOREGRESSIVE MODEL

Abstract. The simultaneous switching autoregressive (SSAR) model proposed by Kunitomo and Sato (A non-linearity in economic time series and disequilibrium econometric models. In Theory and Application of Mathematical Statistics (ed. A. Takemura). Tokyo:University of Tokyo Press (in Japanese), 1994; Asymmetry in economic time series and simultaneous switching autoregressive model. Struct. Change Econ. Dyn. , forthcoming (1994).) is a Markovian non-linear time series model. We investigate the finite sample as well as the asymptotic properties of the least squares estimator and the maximum likelihood (ML) estimator. Due to a specific simultaneity involved in the SSAR model, the least squares estimator is badly biased. However, the ML estimator under the assumption of Gaussian disturbances gives reasonable estimates.     

A similarity‐based approach to time‐varying coefficient non‐stationary autoregression

We suggest in this article a similarity‐based approach to time‐varying coefficient non‐stationary autoregression. In a given sample, the model can display characteristics consistent with stationary, unit root and explosive behaviour, depending on the similarity between the dependent variable and its past values. We establish consistency of the quasi‐maximum likelihood estimator of the model, with a general norming factor. Asymptotic score‐based hypothesis tests are derived. The model is applied to a data set comprised of dual stocks traded in NASDAQ and the Tokyo Stock Exchange.     

Mixtures of Nonlinear Poisson Autoregressions

We study nonlinear infinite order Markov switching integer‐valued ARCH models for count time series data. Markov switching models take into account complex dynamics and can deal with several stylistic facts of count data including proper modelling of nonlinearities, overdispersion and outliers. We study structural properties of those models. Under mild conditions, we prove consistency and asymptotic normality of the maximum likelihood estimator for the case of finite order autoregression. In addition, we give conditions which imply that the marginal likelihood ratio test, for testing the number of regimes, converges to a Gaussian process. This result enables us to prove that the BIC provides a consistent estimator for selecting the true number of regimes. A real data example illustrates the methodology and compares this approach with alternative methods.     

Quasi‐Maximum Likelihood Estimation for a Class of Continuous‐time Long‐memory Processes

Tsai and Chan (2003) has recently introduced the Continuous‐time Auto‐Regressive Fractionally Integrated Moving‐Average (CARFIMA) models useful for studying long‐memory data. We consider the estimation of the CARFIMA models with discrete‐time data by maximizing the Whittle likelihood. We show that the quasi‐maximum likelihood estimator is asymptotically normal and efficient. Finite‐sample properties of the quasi‐maximum likelihood estimator and those of the exact maximum likelihood estimator are compared by simulations. Simulations suggest that for finite samples, the quasi‐maximum likelihood estimator of the Hurst parameter is less biased but more variable than the exact maximum likelihood estimator. We illustrate the method with a real application.     

ON THE UNIMODALITY OF THE EXACT LIKELIHOOD FUNCTION FOR NORMAL AR(2) SERIES

Abstract. For stationary second-order autoregressive normal processes, the conjecture of uniqueness of the solution of the exact likelihood equations is examined. A sufficient condition for uniqueness is given; this condition is satisfied with very high probability if the number of observations is not extremely small. Moreover, it is shown that not more than two maxima may exist. Examples of data which actually produce a likelihood function with two local maxima are given.     

Maximum Likelihood Estimation for a First‐Order Bifurcating Autoregressive Process with Exponential Errors

Abstract. Exact and asymptotic distributions of the maximum likelihood estimator of the autoregressive parameter in a first‐order bifurcating autoregressive process with exponential innovations are derived. The limit distributions for the stationary, critical and explosive cases are unified via a single pivot using a random normalization. The pivot is shown to be asymptotically exponential for all values of the autoregressive parameter.     

Robust estimation for the covariance matrix of multi‐variate time series

In this article, we study the robust estimation for the covariance matrix of stationary multi‐variate time series. As a robust estimator, we propose to use a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (1998) . Particularly, the MDPDE is designed to perform properly when the time series is Gaussian. As a special case, we consider the robust estimator for the autocovariance function of univariate stationary time series. It is shown that the MDPDE is strongly consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.